$\xi R \phi^2$ coupling, cosmological constant and quantum gravitational correction to Newton's potential

This letter investigates the contribution of the $\sqrt{-g}\xi R\phi^2$ interaction to the long range gravitational potential for massive scalar fields, from the non-relativistic limit of the 2-2 scattering amplitude with graviton exchanges. Such coupling is naturally motivated from the renormalisation of a scalar field theory with quartic self interaction in a curved spacetime. This is qualitatively different from the minimal ones like $\sqrt{G} h^{\mu\nu}T_{\mu\nu}$, as the vertices corresponding to the former do not explicitly contain any scalar momenta, but instead explicitly contains the momentum carried by graviton line. For the minimal vertex, the long range gravitational potential up to one loop $({\cal O}(G), {\cal O}(G^2))$ was obtained earlier from the terms non-analytic in the transfer momentum, $q^{-2},\ q^{-1},\ \ln q^2$, yielding potentials respectively like $r^{-1}$, $r^{-2}$, $r^{-3}$. However owing to the aforesaid explicit appearance of transfer momentum for the non-minimal vertices, the leading contribution in this case comes at ${\cal O}(\xi G^2)$, and turns out to be subleading compared to even $r^{-3}$. To complement this `screening' effect, we consider the three graviton vertex generated by the $\sim \Lambda \sqrt{-g}/G$ term in the action, where $\Lambda$ is the cosmological constant. This vertex does not explicitly contain any graviton momentum. With this vertex, and assuming short scale scattering much small compared to the Hubble horizon, we compute the seagull, the vacuum polarisation and the fish diagrams and obtain the 2-2 scattering amplitudes. The leading gravitational potential at ${\cal O}(\xi \Lambda G^2 )$ behaves like $r^{-1}$, even though it is much subleading compared to Newton's potential due to the appearance of $\Lambda$. We also discuss the scenario where this potential dominates the aforesaid ${\cal O}(\xi G^2)$ one.